Relational GNNs Cannot Learn $C_2$ Features for Planning

TL;DR

This paper demonstrates that Relational GNNs are unable to learn $C_2$ features for planning, challenging previous assumptions, and suggests alternative architectures that might succeed.

Contribution

It provides a theoretical and empirical analysis showing R-GNNs cannot learn $C_2$ features, and identifies potential architectures better suited for this task.

Findings

R-GNNs cannot learn $C_2$ features for planning.

Prior GNN architectures may better learn $C_2$ features.

Contradicts empirical results suggesting R-GNNs can learn these features.

Abstract

Relational Graph Neural Networks (R-GNNs) are a GNN-based approach for learning value functions that can generalise to unseen problems from a given planning domain. R-GNNs were theoretically motivated by the well known connection between the expressive power of GNNs and $C_2$, first-order logic with two variables and counting. In the context of planning, $C_2$ features refer to the set of formulae in $C_2$ with relations defined by the unary and binary predicates of a planning domain. Some planning domains exhibit optimal value functions that can be decomposed as arithmetic expressions of $C_2$ features. We show that, contrary to empirical results, R-GNNs cannot learn value functions defined by $C_2$ features. We also identify prior GNN architectures for planning that may better learn value functions defined by $C_2$ features.